$\Omega(\mathbb{Z} \times BGL_{\infty}) \simeq GL_{\infty}$?

Question

Could somebody help to explain the homotopy equivalence $\Omega(\mathbb{Z} \times BGL_{\infty}) \simeq GL_{\infty}$? Dan Freed in his notes pp.7 says it is trivially true but I can't see it.

Answer 1

Just to close the question, which was answered in the comments: $\Omega(\mathbb{Z}\times BG)=\Omega(BG)$ for groups $G$, as the based loop space functor only depends on the pathconnected component containing the basepoints. More or less by definitition $\Omega(BG)\cong G$, which implies the result of the question.